1 Introduction

Traditional water resource allocation usually regards the maximization of social and economic benefits as the primary objectives and ignores ecological benefits, which leads to unrestrained and one-sided development of the social economy and eventually results in a vicious circle of ecosystem degradation. In view of this phenomenon, Wu Meimei et al. investigated the evaluation index system of urban water resource utilization benefits and examined the system under the conditions that the relative importance of indices is determined and the exact weights are unknown (Wu et al. 2020). Qin Jianan et al. also investigated the benefit compensation mechanism in water resource management, and a Stackelberg game theoretical model was used to derive the agent disagreement utility based on consideration of their spatial heterogeneity in terms of water accessibility (Qin et al. 2020). In addition, with an increased focus on ecological benefits, studies on sewage collection and reuse of reclaimed water have been conducted (Castañer et al. 2020; Dehaghi and Khoshfetrat 2020). However, these problems should be gradually extended to the field of water resource allocation.

In the new century, with the progress and improvement of intelligent optimization algorithms, such as immune genetic algorithms and neural network algorithms, these algorithms have been widely applied in the field of water resource allocation. Scholars have also investigated the mechanism of water resource allocation in a broader field to essentially resolve the contradiction of water resource allocation (Perera et al. 2005; Sethi et al. 2006; Abolpour et al. 2007; Zaman et al. 2009; de Lange et al. 2010). Simultaneously, with the development of equilibrium theory, game theory and other theories, the development of socioeconomic models and hydrological models has opened up a new field for the development of water resource allocation modelling systems. Minsker et al. (2000) constructed a multi-objective water resource allocation model based on a genetic algorithm through the uncertainty of hydrological elements and vividly described the uncertain factors in the water resource allocation process. Rosegrant et al. (2000) constructed a hydrologic-economic coupling model for evaluating the efficiency of water resource utilization. Mahan et al. (2002) and Kralisch et al. (2003) applied the neural network algorithm to various water source allocation problems. Kucukmehmetoglu (2012) constructed a model for solving the problem of water resource allocation in cross-border basins, which coupled game theory and Pareto optimal theory to expand water resource allocation to the field of national macro-control. Overall, the studies on water resource allocation started from a single subject, with a single objective as the starting point, and gradually expanded to multiple fields and developed into the current stage of integrating multiple subjects to solve multiple objectives (Nasiri-Gheidari et al. 2018; Pourmand and Mahjouri 2018; Kicsiny and Varga 2019).

Research on water resource allocation in China has entered the stage of harmonious development between humans and water resources. In this stage, sustainable development should be regarded as the starting point for protecting the economic benefits of water resources and emphasizing the water resource attributes and ecological environmental benefit to realize balanced development between humans and the water environment and promote the sustainable development of human society. Wang et al. (2003) proposed the “natural-artificial” dual water cycle theory, which innovatively integrated the water cycle process into the process of social and economic development and injected new ideas into the study of water resource allocation. Xie et al. (2002) fully considered the carrying capacity of water resources and developed water resource allocation models that are suitable for sustainable economic and social development in Ningxia and Xinjiang. Subsequently, the theory of “cubic equilibrium” allocation (Wang et al. 2003) was proposed, which further promoted the expansion of the water resource allocation field. During this period, the water resource allocation model, in combination with the concept of sustainable development, was practically applied in various regions (Chen et al. 2002; Xie et al. 2002). This marked the gradual establishment of the water resource allocation stage of harmonious development between human beings and the water environment. Zhao (2006) and Pei et al. (2007) regarded generalized water resources as the allocation object, the water quantity and quality requirements of each water use unit as the control index, and the “natal-artificial” dual water cycle simulation as the driving factor and realized the sustainable development of the population, water resources and ecological environment in the basin through an optimized decision-making mechanism. Wei et al. (2012) constructed the total factor optimization water resource allocation model based on multiple water resource attributes from a “artificial - natural” binary water cycle theory system, fully considered the regional economic development and water requirement characteristics of each social economic sector, and systematically expounded the theory of total factor optimization allocation of water resource system. The practical application of water resource allocation based on the harmonious development of human and water resources largely coordinates the relationship between socioeconomic development and the sustainable utilization of water resources, alleviates the contradiction between water supply and demand in the process of sustainable development, and makes an important contribution to the improvement of the ecological environment (Liu et al. 2010; Li et al. 2018).

Nowadays, the harmonious development of human and water resources and the solution of multiple objectives has been considered in water resource allocation (Kucukmehmetoglu 2012; Mianabadi et al. 2014; Das et al. 2015; Chang et al. 2016; Zeng et al. 2017; Tian et al. 2019; Wang et al. 2019; Dadmand et al. 2020; Li et al. 2020; Pourmand et al. 2020; Sarband et al. 2020). To guarantee harmonious development with nature and realize a socioeconomic development mode that matches the carrying capacity of water resources, the scarcity of water resources and their ecological value should be further considered in the process of water resource allocation. The objective of this study is to propose a set of model calculation methods with three main synergetic allocations, namely, ‘synergetic water supply from various water sources in the water resource subsystem’, ‘synergetic water requirements among the social, economic and ecological subsystems’ and ‘synergetic balance between the water supply in the water resource subsystem and the water demands in the social, economic and ecological subsystems’, which are subject to five total amount controls, namely, ‘total water use control’, ‘total water consumption’, ‘total groundwater intake’, ‘total reclaimed water supply’ and ‘total ecological water demand in the channel’, for the realization of water resource synergetic allocation.

2 Methodology

2.1 Modelling Principle

2.1.1 Internal Relationships in the Composite System

The internal relationships in the composite system are illustrated in Fig. 1. As the resource provider, the water resource subsystem plays an important role in the composite system. In the social subsystem and economic subsystem, first, it is necessary to judge whether the domestic water demand and production water demand match the water resource carrying capacity. If yes, proceed to the second judgement of whether the production and domestic sewage meet the standards. In the ecological subsystem, the most important principle is to ensure that the inner-river water quantity can satisfy the inner-river ecological water demand. The next step is to judge whether the urban ecological water demand can be satisfied. If yes, proceed to the last judgement of whether the recycled water rates, which correspond to the recycled water that is used in three subsystems, meet the requirement. If yes, the synergetic allocation of water resources can be conducted.

Fig. 1
figure 1

Internal relationships in the composite system

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2.1.2 Synergetic Water Demand Module that Is Based on the Benefit Hierarchy Obedience Principle

The relationship between synergetic theory and water resource allocation in this study is illustrated in Fig. 2. Sometimes, human activities do not lead to the synergy of economic, social and ecological benefits, and it is necessary to adjust the order of economic, social and ecological benefits according to the regional resources, the environmental carrying capacity, and the development demand to formulate a hierarchical development plan (Guo and Shen 1991; Gui 2001). In this study, the study area was divided into key development zones, restricted development zones and strictly controlled development zones.

Fig. 2
figure 2

Synergetic theory and water resource allocation

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In the key development zones, economic benefits and social benefits should be given equal attention, and ecological benefits should be subordinated to social and economic benefits. Hence, in areas with abundant resources and large environmental carrying capacities, it is equally important to guarantee the domestic and industrial water supplies, and the water supply for the ecological environment can be suitably reduced. In restricted development zones, economic benefits should be of the same importance as ecological benefits and subordinate to social benefits, namely, in areas with low resources and environmental carrying capacities, the domestic water supply should be guaranteed first, and production and ecological water use should be maintained cooperatively. In strictly controlled development zones, ecological benefits should be subordinated to social benefits and economic benefits to ecological benefits. Hence, in areas with low resource and environmental carrying capacities, the domestic water supply should be prioritized, water ecological protection should be strictly implemented, and the water supply for economic development should be restricted if necessary.

2.1.3 Synergetic Water Supply Module that Is Based on the System Comprehensive Effect Law

The combination of established and planned water supply projects and water sources will be optimized and the available water supply from various water sources will be analysed according to the regional water resource conditions and water demand structure to provide a basis for the water supply constraint of water resource synergetic allocation. The water supply efficiency will be maximized through the orderly combination and synergetic allocation of various water sources (Guo and Shen 1991; Gui 2001).

From the perspective of water resource quality, water sources with satisfactory water quality should be prioritized domestically to realize higher water supply benefits. The reclaimed water supply to industries with low water quality requirements should be increased, which can yield not only water-saving benefits but also ecological benefits by reducing pollution sources. In terms of water resource quantity, the areas with abundant surface water should be prioritized for surface water supply, and groundwater resources can be regarded as strategic reserve water sources. In areas with limited surface water resources but with groundwater as the main source, external water diversion projects should be planned and implemented, and the local groundwater and limited surface water should be regarded as strategic reserve water sources. From the perspective of water demand, it is necessary to increase the recycling rate of water resources and enhance the reclaimed water quality to satisfy irrigation water quality requirements in industrial and agricultural areas. Meanwhile, the reclaimed water supply to the urban ecological environment should be increased to conserve conventional water resources and guarantee water supply security.

2.1.4 Synergetic Degree Analysis Module

The synergetic degree analysis module is used mainly to evaluate and analyse the synergetic degrees of social, economic, and ecological subsystems based on water resource allocation schemes and recommend a water resource allocation scheme with an optimal synergetic degree for policymakers (Liu and Chen 2009; Lei et al. 2017).

Order Parameter Selection in each Subsystem

In this paper, two order parameters are selected for each subsystem and are presented in Table 1. The social benefits mainly reflect the degrees to which the water supply satisfies the population and economic development demands. Therefore, the per capita water supply and the comprehensive Gini coefficient of the water supply are selected as the order parameters to reflect the social benefits (Liqin et al. 2015; Dai et al. 2018).

$${Gini}_j=1-\sum\limits_{i = 1}^{n} \left({X}_i-{X}_{i-1}\right)\left({Y}_i+{Y}_{i-1}\right)$$
(1)
$$Gini={\lambda}_1{Gini}_1+{\lambda}_2{Gini}_2+{\lambda}_3{Gini}_3$$
(2)

where Ginij is the Gini coefficient of subitem j (j = 1,2,3); Xi represents the cumulative percentage of population, GDP, and water resources in the ith administrative region; Yi represents the cumulative percentage of water consumption in the ith administrative region; and λ1, λ2, and λ3 represent the influencing weight coefficients on the water consumption allocation fairness of each sub-Gini coefficient, which satisfy λ1 + λ2 + λ3 = 1. According to the equal subsystem importance law of synergetic theory, λ1 = λ2 = λ3 = 1/3. The economic benefit is mainly reflected by the relationship between the water supply and water deficit. The water supply of each industry is selected as the positive order parameter of the economic subsystem, and the total water deficit is selected as the reverse order parameter of the economic subsystem. The order degree of the ecological subsystem is affected mainly by water environment problems and ecological environmental protection measures. Therefore, the ecological benefits are increased mainly by reducing the discharge of sewage into rivers and increasing the ecological environment water supply. In this paper, the sewage recycling rate is selected as an order parameter of the ecological subsystem to reflect the ecological benefit of reducing sewage discharge, and the water supply for the ecological environment is selected as another order parameter of the ecological subsystem.

Table 1 Order parameters of the three subsystems
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Calculation of the Order Degrees in the Subsystems

The order degree in a subsystem reflects the order degree of the interaction between subsystem elements, which is calculated via the fuzzy mathematical method. The order parameters in subsystems include positive order parameters, which are assigned the ‘+’ tag in Table 1, and reverse order parameters, which are assigned the ‘-’ tag. The larger the value of the positive order parameter, the higher the order degree of the subsystem. The order degree of the ith positive order parameter eji of the jth subsystem is calculated via formula (3). In contrast, the smaller the reverse order parameter is, the higher the order degree of the system. The order degree of the ith reverse order parameter eji of the jth subsystem is calculated via formula (4) (Yao et al. 2017).

$${F}_j\left({e}_{ji}\right)=\frac{e_{ji} {‐} {\beta}_{ji}}{\alpha_{ji} {‐} {\beta}_{ji}}$$
(3)
$${F}_j\left({e}_{ji}\right)=\frac{\alpha_{ji} {‐} {e}_{ji}}{\alpha_{ji} {‐} {\beta}_{ji}}$$
(4)

where αji and βji are the threshold values of the ith-order parameter of the jth subsystem and βji ≤ eji ≤ αji. The value of each order parameter is between 0 and 1, and the larger the Fj(eji) value is, the greater its contribution to the order degree of its subsystem. The order degree Fj(ej) of the jth subsystem is calculated via formula (5), where λi is the influencing weight coefficient of the ith-order parameter of the subsystem.

$$\left\{\begin{array}{*{20}c}F_j\left({e}_j\right)=\sum\limits_{i = 1}^{n}{\lambda}_i{F}_j\left({e}_{ji}\right)\\ {}{\lambda}_i>0\\ \sum\limits_{i=1}^{n}{\lambda}_i=1 \\ \end{array} \right.$$
(5)

Synergetic Degree Calculation of Water Resource Allocation Schemes

The synergetic degree of a water resource subsystem is affected by the order degrees of the social, economic and ecological subsystems, and the synergetic degree is calculated via formula (6), where γj is the influencing weight coefficient of the order degree of the jth subsystem.

$$\left\{\begin{array}{*{20}c}D=\sum \limits_{j=1}^k{\gamma}_j{F}_j\left({e}_j\right)\\ {}{\gamma}_j>0\\ {}\sum \limits_{j=1}^k{\gamma}_j=1\end{array}\right.$$
(6)

2.2 Modelling Specification

2.2.1 Objective Functions of Synergetic Allocation Based on the Optimal System Comprehensive Benefit Law

In the process of water resource allocation, the optimal comprehensive benefit is regarded as the objective, which includes the benefit objectives of the social, economic and ecological subsystems, which are denoted by F1, F2, and F3, respectively, and the comprehensive benefit of water resource allocation is maximized by optimally combining the benefits of the three subsystems. The objective functions are presented as formula (7) to formula (15). The variables and parameters in these formulas are defined in Tables 2 and 3.

Table 2 Definitions of the variables
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Table 3 Definitions of the parameters
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The social benefit objective is reflected mainly by the security level of the water supply. Multisource synergetic water supply for various industries should be conducted in each allocation unit. Reasonable weight coefficients for the water supply from various water sources should be set according to the water source conditions and water demands of the industries to determine the optimal water security level.

$$F_{1} = Max\left( {f_{C} + f_{1} + f_{A} + f_{E} + f_{R} } \right)$$
(7)
$$f_{C} = \sum\limits_{i = 1}^{m} {a_{i}^{C} } \cdot \sum\limits_{j = 1}^{c} {\left( {\alpha_{j}^{sur - C} \cdot S^{C}_{ij} + \alpha_{j}^{gra - C} \cdot G^{C}_{ij} + \alpha_{j}^{slf - C} \cdot SF^{C}_{ij} + \alpha_{j}^{div - C} \cdot D^{C} ij} \right)}$$
(8)
$$f_{R} = \sum\limits_{i = 1}^{m} {a_{i}^{R} } \cdot \sum\limits_{j = 1}^{c} {\left( {\alpha_{j}^{sur - R} \cdot S^{R}_{ij} + \alpha_{j}^{gra - R} \cdot G^{R}_{ij} + \alpha_{j}^{slf - R} \cdot SF^{R}_{ij} + \alpha_{j}^{div - R} \cdot D^{R} ij} \right)}$$
(9)
$$f_{A} = \sum\limits_{i = 1}^{m} {a_{i}^{A} } \cdot \sum\limits_{j = 1}^{c} {\left( {\alpha_{j}^{sur - A} \cdot S^{A}_{ij} + \alpha_{j}^{gra - A} \cdot G^{A}_{ij} + \alpha_{j}^{slf - A} \cdot SF^{A}_{ij} + \alpha_{j}^{div - A} \cdot D^{A} ij + \alpha_{j}^{rec - A} \cdot T^{A} ij} \right)}$$
(10)
$$f_{I} = \sum\limits_{i = 1}^{m} {a_{i}^{I} } \cdot \sum\limits_{j = 1}^{c} {\left( {\alpha_{j}^{sur - I} \cdot S^{I}_{ij} + \alpha_{j}^{gra - I} \cdot G^{I}_{ij} + \alpha_{j}^{slf - I} \cdot SF^{I}_{ij} + \alpha_{j}^{div - I} \cdot D^{I} ij + \alpha_{j}^{rec - I} \cdot T^{I} ij} \right)}$$
(11)
$$f_{E} = \sum\limits_{i = 1}^{m} {a_{i}^{E} } \cdot \sum\limits_{j = 1}^{c} {\left( {\alpha_{j}^{sur - E} \cdot S^{E}_{ij} + \alpha_{j}^{gra - E} \cdot G^{E}_{ij} + \alpha_{j}^{slf - E} \cdot SF^{E}_{ij} + \alpha_{j}^{div - E} \cdot D^{E} ij + \alpha_{j}^{rec - E} \cdot T^{E} ij} \right)}$$
(12)

where fC, fI, fA, fE, and fR are the water supply benefits of urban life, industry, agriculture, ecology and rural life, respectively; \({\alpha}_i^C\), \({\alpha}_i^I\), \({\alpha}_i^A\), \({\alpha}_i^A\), and \({\alpha}_i^A\) are the weights of the water supply for urban life, industry, agriculture, ecology and rural life, respectively, in the ith water source area; \({\alpha}_j^{sur-C\left(I,A,E,R\right)}\),\({\alpha}_j^{gra-C\left(I,A,E,R\right)}\),\({\alpha}_j^{slf-C\left(I,A,E,R\right)}\), and \(\alpha_{j}^{{div - C\left( {I,A,E,R} \right)}}\) are the weight coefficients of the river surface water, groundwater, local surface water and diverted water supply for urban life, industry, agriculture, ecology and rural life, respectively, of the jth calculation unit; and \({\alpha}_j^{re\mathrm{c}-I\left(A,E\right)}\) is the weight coefficient of the reclaimed water supply for agriculture and ecology of the jth calculation unit.

The economic benefit objective is reflected by the water deficit. For industries, the smaller the water deficit is, the higher the economic benefit of the water supply. Each industry should be assigned a corresponding water deficit weight coefficient according to the industry structure and water requirements in the allocation units to minimize the area water deficit and realize synergetic economic development.

$$F_{2} = Min\sum\limits_{j = 1}^{n} {\left( {\beta_{j}^{C} \cdot M^{C}_{j} + \beta_{j}^{I} \cdot M^{I}_{j} + \beta_{j}^{A} \cdot M^{A}_{j} + \beta_{j}^{E} \cdot M^{E}_{j} + \beta_{j}^{R} \cdot M^{R}_{j} } \right)}$$
(13)

where \({\beta}_j^C,{\beta}_j^I,{\beta}_j^A,{\beta}_j^E,{\beta}_j^R\) are the weight coefficients of the urban living water deficit, industrial water deficit, agricultural water deficit, ecological water deficit and rural living water deficit, respectively, of the jth calculation unit.

The ecological benefit objective considers two main factors: the reclaimed water supply for the urban ecological environment and the inner river ecological water supply. The reclaimed water supply for the urban ecological environment is maximized, and the inner river ecological water deficit is minimized. The objective function is expressed as follows:

$$F_{3} = Max\sum\limits_{j = 1}^{n} {\left( {\gamma_{j}^{rec} \cdot T^{R}_{j} - \gamma_{j}^{E} \cdot M^{E} j} \right) + Max\sum\limits_{x = 1}^{k} {\left[ {S_{x} \left( t \right)} / {D_{x} \left( t \right)} \right]} }$$
(14)

where \({\gamma}_j^{rec}\) and \({\gamma}_j^E\) are the weight coefficients of the reclaimed water supply for the urban ecological environment and the water deficit for the ecological environment, respectively, of the jth calculation unit, and Sx(t) and Dx(t) are the inner river water supply and the inner river ecological water demand, respectively, of the xth reach in time period t.

The final objective is to realize the optimal comprehensive benefit of water resource allocation. The water resources and ecological environment carrying capacity are comprehensively considered in the allocation units according to the water supply benefit hierarchy obedience principle, and reasonable weight coefficients are assigned to the water supply benefit objectives of the subsystems.

$$F={\mu}_1{F}_1+{\mu}_2{F}_2+{\mu}_3{F}_3$$
(15)

where F is the final objective and μ1, μ2, and μ3 are the weight coefficients of the social objective, economic objective and ecological objective, respectively.

2.2.2 Constraint Conditions

The constraint conditions consist mainly of available water supply constraints, economic benefit constraints and ecological benefit constraints. The available water supply constraints include the local available water supply constraint, groundwater supply constraint, diverted water supply constraint and reclaimed water supply constraint, as expressed in formulas (16) to (19).

$${SF^C}_{tm}^j+{SF^I}_{tm}^j+{SF^A}_{tm}^j+{SF^E}_{tm}^j+{SF^R}_{tm}^j\le {SF}^{P- Cj}\cdot {SF^P}_{tm}^j$$
(16)
$${G^C}_{tm}^j+{G^I}_{tm}^j+{G^A}_{tm}^j+{G^E}_{tm}^j+{G^R}_{tm}^j\le {G}^{P- Uj}\cdot {G^Q}_{tm}^j$$
(17)
$${D^C}_{tm}^j+{D^I}_{tm}^j+{D^A}_{tm}^j+{D^E}_{tm}^j+{D^R}_{tm}^j\le {D^Q}_{tm}^j$$
(18)
$${T^I}_{tm}^j+{T^A}_{tm}^j+{T^E}_{tm}^j\le {T^Q}_{tm}^j$$
(19)

The economic benefit constraints include mainly the benefits per cubic metre water supply for industry and GDP, as expressed in formula (20).

$$\left\{\begin{array}{*{20}c}\sum\limits_{j = 1}^{n}{I}^j\ge {I}_s\\ \sum\limits_{j = 1}^{n}{G}^j\ge {G}_s\end{array}\right.$$
(20)

where I j and G j are the per cubic metre water benefit values of industry and GDP, respectively, of the jth calculation unit, and Is and Gs are the planned target per cubic metre water benefits of industry and GDP, respectively.

The ecological benefit constraints include mainly sewage reuse constraints of the calculation unit and ecological constraints of river courses, as expressed in formulas (21)–(23), where λj is the planned sewage recycling rate of the jth calculation unit.

$${T^R}_{tm}^j=\left({W^C}_{tm}^j-{M^C}_{tm}^j\right)\cdot {S^{P-C}}_{tm}^j\cdot {T^{P- CD}}_{tm}^j\cdot {T^{P- CT}}_{tm}^j\cdot {T^{P- CR}}_{tm}^j+\left({W^I}_{tm}^j-{M^I}_{tm}^j\right)\cdot {S^{P-I}}_{tm}^j\cdot {T^{P- ID}}_{tm}^j\cdot {T^{P- IT}}_{tm}^j\cdot {T^{P- IR}}_{tm}^j$$
(21)
$$\sum \limits_{j=1}^n{T^R}_{tm}^j\ge {\lambda}^j\cdot {T^S}_{tm}^j$$
(22)
$${R}_{\mathrm{max}}^l\ge {R}^l\ge {R}_{\mathrm{min}}^l$$
(23)

2.2.3 Balance Equation

The balance equations of the water allocation model in the calculation units are presented as formulas (24) to (28).

$${W^C}_{tm}^j={SF^C}_{tm}^j+{S^C}_{tm}^j+{D^C}_{tm}^j+{P^C}_{tm}^j+{G^C}_{tm}^j+{M^C}_{tm}^j$$
(24)
$${W^I}_{tm}^j={SF^I}_{tm}^j+{S^I}_{tm}^j+{D^I}_{tm}^j+{P^I}_{tm}^j+{T^I}_{tm}^j+{G^I}_{tm}^j+{M^I}_{tm}^j$$
(25)
$${W^A}_{tm}^j={SF^A}_{tm}^j+{S^A}_{tm}^j+{D^A}_{tm}^j+{P^A}_{tm}^j+{T^A}_{tm}^j+{SN^A}_{tm}^j+{G^A}_{tm}^j+{M^A}_{tm}^j$$
(26)
$${W^R}_{tm}^j={SF^R}_{tm}^j+{S^R}_{tm}^j+{D^R}_{tm}^j+{P^R}_{tm}^j+{G^R}_{tm}^j+{M^R}_{tm}^j$$
(27)
$${W^E}_{tm}^j={SF^E}_{tm}^j+{S^E}_{tm}^j+{D^E}_{tm}^j+{P^E}_{tm}^j+{T^E}_{tm}^j+{G^E}_{tm}^j+{M^E}_{tm}^j$$
(28)

2.3 Multiple Iteration Algorithm of the Model

The technical process of the model includes three main layers: the basic system layer, the input layer and the allocation scheme generation layer. The calculation processes are illustrated in Fig. 3.

Fig. 3
figure 3

Multiple iteration algorithm of the model

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3 Case Study

3.1 Study Area and Data

Jilin City is located in east Jilin Province. The main river of this city is the Songhua River, which has a watershed area of 15,120 km2 and a river length of 432 km within the territory of Jilin city. In addition to the trunk stream of the Songhua River and its main tributaries of the Huifa River and Yinma River, there are 9 tributaries with watershed areas of more than 500 km2, which are presented in Fig. 4. Moreover, Jilin has abundant rainfall and a developed urban water system. The abundant water resources and various types of water supply and drainage projects can fully reflect the diversity and synergy of water sources and water supply projects. Additionally, the industry structure of Jilin is gradually transforming from agriculture and heavy industry to light manufacturing and service industries, which is representative of the industry structure of China. Finally, the spatial distribution of water resources in Jilin is uneven, and the contradictions between water supply and water demand differ among parts of this city, which can fully reflect the diversity of contradictions between socioeconomic development and environmental carrying capacity.

Fig. 4
figure 4

River system in Jilin

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In this study, long-series groundwater data from 2006 to 2016 and long-series channel and project node runoff data from 1956 to 2016 were collected from the Hydrographic Office of Jilin. The drainage data, water supply data and the scales of water conservancy projects were collected from the Water Conservancy Bureau of Jilin. To reduce the error that was caused by the uncertainty regarding the agricultural and ecological water demands, long-series monthly data were used to simulate the water supply and demand configuration.

3.2 Regional Development Hierarchy Positioning

According to the benefit hierarchy obedience principle, the positioning of regional development levels is determined, which is presented in Table 4. The first development level is the key development zone, where the urbanization and industrialization development speed exceeds those in other areas of the city. The second development level is the restricted development zone, where the urbanization and industrialization development speed is lower than that of the first level. The third development level is the strict control zone, where the urbanization and industrialization development speed is the lowest among the three levels.

Table 4 Positioning of the regional development levels
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3.3 Water Resource Allocation Schemes

In this study, two development models of high-speed development and moderate development were proposed. The social and economic indicators were predicted based on the indicators in the base year. Two water-saving plans for strengthening water savings and moderate water savings were also proposed. The water use efficiency indices were predicted based on the indices in the base year. The growth plans of social and economic indicators were combined with the water-saving plans to generate four water demand prediction schemes, which are denoted as Scheme I, Scheme II, Scheme III and Scheme V and are presented in Table 5. Since the predicted result of Scheme V is similar to that of Scheme I and high-speed development with the moderate water saving scheme better conforms to the requirements of sustainable development, Scheme V was abandoned. Scheme I, Scheme II and Scheme III were designated as the high-water-demand scheme, the medium-water-demand scheme and the low-water-demand scheme, respectively.

Table 5 Water demand schemes
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3.4 Construction of the Water Resource Allocation Network

A network diagram of the water resource allocation system is presented in which is Fig. 5. It illustrates the topological relationships among calculation units, important water conservancy projects, various types of water flow transmission systems, and important control sections and nodes.

Fig. 5
figure 5

Water resource allocation network diagram of Jilin

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3.5 Determination of the Order Parameter Thresholds of the Subsystems

In this paper, the critical values are determined according to the constraint conditions of each order parameter. In the social subsystem, for the per capita water supply, 440 m3 of the national per capita water supply was selected as the lower threshold, and the per capita available water supply was selected as the upper threshold. For the comprehensive Gini coefficient of the water supply, we selected 0 ~ 0.4 as its critical threshold to ensure that the water resource allocation and socioeconomic development can at least realize a relative matching level according to the value standard of the Gini coefficient, which is presented in Table 6. In the economic subsystem, for the water supply, the actual water consumption in the base year was selected as the lower limit, and the available water supply in the planning level year was selected as the upper limit. For the water deficit, the full water supply with a water deficit of 0 was selected as the lower limit, and the annual water deficit in special dry years was selected as the upper limit. In the ecological subsystem, 20% and 35% were selected as the lower limit and the upper limit, respectively, of the sewage recycling rate according to the conditions in this study area. For the ecological water supply, the ecological water consumption in the base year was selected as the lower limit, and the ecological water demand in the planning lever year, which could fully satisfy the ecological water demand, was selected as the upper limit.

Table 6 Value standard of the Gini coefficient
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4 Results and Discussion

4.1 Synergetic Water Demand Schemes that Are Based on the Benefit Hierarchy Compliance Principle

As illustrated in Fig. 6, in the areas that were designated as the first level, the GDP and population growth rates of both the high-speed and moderate development models exceed those in the areas of the second level and the third level.

Fig. 6
figure 6

Hierarchically predicted average annual growth rate. (a) GDP and (b) Population

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Moreover, to increase the water-use efficiency, the water demand quotas for a GDP of ten thousand yuan, domestic and agricultural sectors are also predicted hierarchically, as illustrated in Fig. 7. For both the moderate water saving mode and the strengthening water saving mode, the water demand quotas of the 1st level areas exceed those of the 2nd level areas, and those of the 3rd level areas are the lowest. The prediction results for water demand are presented in Table 7.

Fig. 7
figure 7

Prediction of the water demand quota. (a) Ten thousand yuan GDP; (b) domestic; and (c) agriculture

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Table 7 Prediction results of the water demand/104 m3
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The water demand structure of each water demand scheme has the same variation trend from the base year to 2030, which is shown in Fig. 8. In the base year, agriculture accounts for the largest proportion of water consumption, followed by industry. The service industry and ecological environment account for lower proportions. In 2020 and 2030, the industrial, construction, service, domestic, and ecological water demands will gradually increase with the development of urbanization and industrialization. Meanwhile, with the development of water-saving irrigation technology and ecological agriculture, the water demand of agriculture will show a decreasing trend.

Fig. 8
figure 8

Prediction of the water demand structure

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4.2 Synergetic Water Supply Schemes that Are Based on the System Synthesis Effect Law

The main water supply source of Jilin in the base year is conventional water, such as surface water and groundwater. However, reclaimed water is underutilized. To maintain ecological benefits, the utilization of reclaimed water should be increased in the future to realize a synergetic water supply from multiple water sources. By 2020, a reduction in groundwater withdrawal is planned along with an increase in the reclaimed water supply. Meanwhile, the available water supplies of the established water supply projects and the planned water supply projects can satisfy the three water demand schemes in 2020. Therefore, no additional water supply project planning is conducted in 2020 in this study. By 2030, strategic water reserves must be available to match the development model with the carrying capacity of resources and the environment in various regions. In addition to reducing groundwater withdrawal, the establishment of new water storage projects, diversion projects, pumping projects and reclaimed water plants is proposed.

As shown in Fig. 9, available water supply schemes with various water supply guarantee rates are predicted. With the closure of self-contained wells, the available groundwater supply is gradually reduced from the base year to 2030. However, the available reclaimed water supply and available water supply from water storage engineering, pumping engineering and diversion engineering are gradually increasing. In 2030, the commissioning of external diversion engineering will be necessary to meet the increasing water demand.

Fig. 9
figure 9

Prediction results of the available water supply

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4.3 Water Resource Synergetic Allocation Schemes with the Optimal System Comprehensive Benefit

The synergetic water allocation schemes illustrated in Figs. 10 and 11 are established under the multiyear average water supply guarantee rate. Among the three allocation schemes, only Scheme III is a low-speed socioeconomic development scheme. To maximize the economic benefits of the external diverted water supply, external diversion engineering only provides water for the high-speed socioeconomic development schemes, namely, Scheme I and Scheme II. The allocation results demonstrate that by 2020, portions of the groundwater and surface water supplies will be replaced with the increased reclaimed water supply. By 2030, the increased reclaimed water supply and externally diverted water supply will result in significant savings in the local surface and groundwater supplies. The water deficit rate will decrease significantly, especially after the implementation of external diversion engineering.

Fig. 10
figure 10

Water resource synergetic allocation schemes

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Fig. 11
figure 11

Water resource synergetic allocation results: (a) Scheme I, (b) Scheme II, and (c) Scheme III

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4.4 Synergetic Degree Analysis and the Recommendation of a Water Resource Allocation Scheme

The synergetic degree analysis results are presented in Table 8. The social subsystem order degree of Scheme III exceeds those of Scheme II and Scheme I, but the economic subsystem and the ecological subsystem order degrees of Scheme III are below those of Scheme II and Scheme I, which do not support ecological civilization construction. T economic subsystem, the economic subsystem order degree of Scheme I and Scheme II are higher than Scheme III, and the two schemes are corresponding to the high-speed development mode, but the ecological subsystem order degree of Scheme I is lower than Scheme II and Scheme III, and Scheme I has a lower water-saving strength; hence, it is not recommended as the optimal scheme. In terms of the ecological subsystem, Scheme II has the highest synergetic degree, which also has the highest order degree of the ecological subsystem. In addition, Scheme II corresponds to strengthening water savings, which can not only guarantee the rapid development of the social economy but also provide high water supply security, and it is the most conducive to the harmonious development of the three subsystems. Therefore, Scheme II is selected as the recommendation, which is presented in Table 9.

Table 8 Synergetic degrees of water resource allocation schemes
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Table 9 Recommended scheme for synergetic water allocation/104m3, %
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5 Conclusion

  1. (1)

    This paper analyzed the relationships between water resource subsystem and the three subsystems of society, economy and ecology. Provided the theoretical basis for synergetic water demand of the three subsystems, synergetic water supply from various water sources and the synergetic allocation of water resources is provided. Synergetic theory, which includes the benefit hierarchy obedience principle and the system comprehensive effect law, was applied to water resource allocation.

  2. (2)

    We constructed a water resource allocation model, which includes a data pre-processing module, a synergetic water demand module that is based on the benefit hierarchy obedience principle, a synergetic water supply module that is based on the system comprehensive effect law, and a synergetic degree analysis module. In addition, we proposed a multicycle iterative algorithm for realizing the overall objective of “harmonious development between human and water resources”, which provides an effective calculation tool for water resource synergetic allocation.

  3. (3)

    Finally, the model was applied in Jilin, and Scheme II was recommended as the optimal scheme through a synergetic degree analysis. The water supply of the conventional water source will be conserved due to an increase in the reclaimed water supply. In addition, after the implementation of an external diversion project by 2030, the amount of groundwater withdrawal will be gradually reduced, and the water deficit rate will be significantly reduced.